Suppose X*X* is the number of the number of tails obtained before the rth*rth* head occurs in a series of independent tosses of a coin having probability of head p*p*. The random variable is said to have the Negative Binomial distribution with parameters r*r* and p*p*. The probability mass function f(x)*f*(*x*) of this distribution is_____.

The random variable \( X \) representing the number of tails before the \( r \)-th head in a series of independent coin tosses follows the **Negative Binomial distribution** with parameters \( r \) and \( p \), where \( p \) is the probability of getting a head on each toss.

The **probability mass function (PMF)** of the Negative Binomial distribution with parameters \( r \) (the number of heads or "successes") and \( p \) (the probability of a head) is given by:

\[
f(x) = \binom{x + r - 1}{r - 1} p^r (1 - p)^x
\]

where:
- \( x \) is the number of tails before the \( r \)-th head,
- \( r \) is the number of heads (successes),
- \( p \) is the probability of getting a head (success),
- \( 1 - p \) is the probability of getting a tail (failure),
- \( \binom{x + r - 1}{r - 1} \) is a binomial coefficient, which counts the number of ways to arrange \( x \) tails and \( r \)-1 heads in \( x + r - 1 \) trials.

### Explanation:
- You need \( r \) heads (successes), and \( x \) tails (failures) occur before the \( r \)-th head.
- The total number of trials required to achieve this is \( x + r \).

### Example:
If you want the probability that you get 3 tails before getting the 4th head, this would follow a Negative Binomial distribution with \( r = 4 \) and some given probability \( p \) of heads.

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Would you like further details or have any other questions?

### Follow-up Questions:
1. How is the binomial coefficient \( \binom{x + r - 1}{r - 1} \) computed?
2. Can you explain the connection between the Negative Binomial distribution and geometric distribution?
3. How does the PMF change if we consider biased vs. unbiased coins?
4. What is the expected number of tails before the \( r \)-th head for a given \( p \)?
5. How can we derive the Negative Binomial distribution from a sequence of Bernoulli trials?

### Tip:
The Negative Binomial distribution generalizes the geometric distribution, which models the number of failures before the first success (i.e., \( r = 1 \)).

Suppose X is the number of tails obtained before the rth head occurs in a series of independent tosses of a coin having probability of head p. The random variable is said to have the Negative Binomial distribution with parameters r and p. The probability mass function f(x) of this distribution is_____.

Discover how to compute the probability mass function for the number of tails before the rth head occurs in a series of independent coin tosses. This problem utilizes the Negative Binomial distribution with parameters r and p, covering core concepts like binomial coefficients and probability theory.

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